The decomposability of additive hereditary properties of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1, . . . ,Pn are properties of graphs, then a (P1, . . . ,Pn)-decomposition of a graph G is a partition E1, . . . , En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i = 1, . . . , n. We define P1⊕· · ·⊕ Pn as the property {G ∈ I : G has a (P1, . . . ,Pn)-decomposition}. A property P is said to be decomposable if there exist non-trivial hereditary properties P1 and P2 such that P = P1 ⊕ P2. We study the decomposability of the well-known properties of graphs Ik, Ok, Wk, Tk, Sk, Dk and Op.